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I am new to special relativity. I don't have a background in physics.
I was reading Brian Green's The Elegant Universe and got hung up on an example.

In fact, Einstein realized that the principle of relativity makes an even grander claim: the laws of physics—whatever they may be—must be absolutely identical for all observers undergoing constant-velocity motion. If George and Gracie are not just floating solo in space, but, rather, are each conducting the same set of experiments in their respective floating space stations, the results they find will be identical.

I must be missing the point of the principle of relativity because I cannot understand why the two observers moving at a constant velocity, which obviously differs between them in order for their motion to quality as relative motion, would observe the same physics.

I have tried numerous videos and introductory reading materials but they never focus on that part of the reasoning.

My questions are:

  1. Would it be true that the two observes would experience the same physics even if their constant velocities differ?
  2. Would that still be true if the velocity of one or the other observer starts approaching the speed of light?
  3. Is there any condition under which they will experience different physics while traveling at constant velocities?

I am sure my mental model is either wrong or incomplete. Thank you all for the help in advance. This lack of understanding has been torturing me for a week and a half now and I could use a pointer in the right direction. Please feel free to share any resources that you think I should read/explore to get a better understanding.

EDIT: Thank you all for the great answers. I am blown away by the fact that all of you have taken the time to provide as many examples and directions as you did. I come from Computer "Science" and while those communities are great, the number and focus of the responses really surprised me. I also apologize for the delay in my response. I was traveling and did not have access to the internet.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – ACuriousMind
    Aug 26, 2021 at 20:16
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    $\begingroup$ Please don't add a new question to an existing one, especially after you've received answers on the original question. These answers will obviously not have been formulated to address your updated question. I recommend you remove your added question, and ask it as a new question with a link to this question. :-) $\endgroup$ Aug 27, 2021 at 8:25

10 Answers 10

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I cannot understand why the two observers moving at a constant velocity, which obviously differs between them in order for their motion to quality as relative motion would observe the same physics.

Since it seems, you are having trouble getting it intuitively, I will try to explain through an example/thought experiment which is used often when teaching relativity.

Imagine you are floating through a spaceship in a GIANT earth-sized ship. Just like how you are floating through space on the ship called earth, (of course ignoring the motion of Earth around the Sun). Let us say, you and your partner start playing a tennis match inside the ship. Now, you would not need to relearn how the ball moves or retrain your reflexes to play in a flying spaceship. All the laws of motion that govern how the ball moves on Earth, would be the same in that spaceship. And you would be able to play tennis just as well on the ship as you do on earth.

Now, if another of your friend went into another spaceship that is moving relative to your ship, they would be able to play tennis in their ship just as well too. So, here, even though you both are moving at different velocities, you both experience the same laws of physics. If the laws of physics were different in one of the ships, then that person would not be able to play tennis like they are used to.

If the above example does not do it for you, then just think of when you travel on a plane flying at hundreds of kilometers per hour. If you pour a cup of water on the plane, do you have to adjust for the fact that the plane is going at such a huge velocity? No, you do not (of course, if there is turbulence or anything else, then it would not be true). But if you assume, there is a uniform velocity of the plane, then the laws all stay the same for you. You are able to pour the water just as comfortably as you do in your dining room. You do not suddenly have to do anything different to adjust to any new laws of motion.

To answer the specific questions you asked:

Would it be true that the two observes would experience the same physics even if their constant velocities differ?

Yes, as long as the 2 observers are not accelerating, they would experience the same physics.

Would that still be true if the velocity of one or the other other observer starts approaching the speed of light?

Yes, that would still be true. In an outsider's frame, the observers might length contract and time dilate, but in the observer's frame, everything will be as normal as before.

Is there any condition under which they will experience different physics while travelling at constant velocities?

No, if they are travelling at constant velocities, in the frame of the same observer, then they will both experience the same laws of physics.

"Will the values they measure for each of the forces be exactly the same?"

Not necessarily. The statement that the laws of physics would be the same is different from the statement that 2 observers would measure the same values for ALL forces.

A very rough and somewhat unreal example to explain. Let us say, a law of physics was that an object travelling at a velocity v experiences a force called the "rapidity" force which is equal to $v * R$, where $R$ is the "rapidity" force constant.

So, if you are travelling by in a spacecraft past me, with relative velocity $v$, then in your frame, any experiment you do to measure the "rapidity" force exerted on me, would measure a value $v*R$ (which you would expect if the law applies).

But, for another observer, travelling by in another spacecraft past me, with relative velocity, $u$, they would measure $u*R$ (which they would also expect if the law applies ).

So, each of you would be satisfied that your measurement verifies the same law of physics. But, the force measured by each of you would be different.
Hence, even though the laws of physics stay the same, the forces measured can be different.

Please feel free to share any resources that you think I should read/explore to get a better understanding.

I feel that at the level that you are at, trying to get into relativity, you would really enjoy the video: Brian Wolfson's "Einstein's Relativity and the Quantum Revolution: Modern Physics for Non-Scientists"

I recommend it to everyone who wants to understand relativity but is just unable to get the intuition behind it. The original video was from this link. I can no longer find the videos on youtube. You can get the book version if you want to read it as well.

You can get the audiobook on audible. I looked it up on youtube and was able to find an audio version of the first one or 2 videos. The video says 6 hours long, but they are on a loop. It's worth a try, to figure out if you like it and whether it is worth trying to get the whole thing.

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    $\begingroup$ The turbulence you feel IS acceleration. We associate getting jostled with speed simply because most of our transportation methods inefficiently bump us around. But that's not a law of physics. It's just less than ideal air, road, or rails. Theoretical ships traveling in space (and earth for that matter) get to travel through an idealized nothing, free of potholes. So you never feel the speed. All you have ever felt has been acceleration. $\endgroup$ Aug 24, 2021 at 18:26
  • $\begingroup$ Thank you for your thorough answer @thinking squares. This really helped me improve how I was viewing it. The main challenge I seem to be having is that I come from a field where laws determine a priori and there is little (with some exceptions) observed unplanned behavior. In Physics, it seems that the approach is to determine a hypothesis that predicts the behavior and verify experimentally. With my questions I was looking for the "why" should this be true which does not seem to make sense in the frame of Physics. $\endgroup$ Aug 25, 2021 at 14:42
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    $\begingroup$ An excellent argument, originating with Galileo. $\endgroup$
    – J.G.
    Aug 25, 2021 at 21:18
  • $\begingroup$ @DimitarDimitrov: You may find this lecture by Feynman helpful. $\endgroup$
    – Kevin
    Aug 26, 2021 at 0:01
  • $\begingroup$ Your "Rapidity" force would be a non-relativistic force, no? $\endgroup$
    – Yakk
    Aug 26, 2021 at 13:32
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The whole point is that this is an observation that is made -- if you're on a windowless train moving on a smooth track, you don't really know you're moving at all.

Similarly, for all of newtonian mechanics, if you go and subsitute a velocity $v$ for a velocity $w = v + c$, where $c$ is a constant velocity vector, all of the laws will simply transfer to laws with $w$ in them, rather than $v$.

This all changed when the final form of Maxwell's equations were worked out. Suddenly, that simple velocity transformation didn't make the laws invariant, because Maxwell's equations predicted a speed for light. Einstein's whole insight was, "this principle of relativity is the fundamental thing", which then meant that all observers needed to see the same speed of light.

The principle of relativity is the assumption put into the theory. it is justified with some simple observations, and its consistency with previous theory, but from the point of view of special relativity, it is simply an axiom.

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    $\begingroup$ thank you for including the bit about "is the assumption put into theory". This helped me start seeing the flaw in how I am approaching these issue. Trying to determine some fundamental reason why a hypothesis/law should be correct may not be best way to look at this. It seems due to my background I am trying to approach these as: Why must this be true? What would break if it is not? That does not seem to be the right approach here. $\endgroup$ Aug 25, 2021 at 14:53
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So this “reference frame equivalence” claim happens way before relativity, relativity just reconciles it with a new observation: a constant speed of light. This is a very strange observation to mix with reference frame equivalence, but Hendrik Lorentz worked out the formula (it is basically a Doppler shift tied to acceleration times distance, added to the usual one tied to velocity) and Einstein argued it was a universal feature of acceleration rather than some weirdness of electromagnetism, and Minkowski embedded it in a 4D hyperbolic space—and this became what we now call “special relativity.” But reference frame equivalence existed before all of that.

So I recommend that you think about it in those terms. At least at first. Don't include the weird Doppler shift that keeps the speed of light constant, just yet.

So what this says is, that I can juggle on a train.

Obviously, I can't always juggle very well on a train, sometimes the tracks are mangled and everybody gets knocked down. Other times, maybe there is a car on the track ahead and the train conductor has pulled the emergency brake and we are all being flung into the wall at the front of the train, which makes it very hard to juggle too. I would also notice if we took a steep turn a little too fast, while I was juggling my balls would be moving in an unusual way through the air as seen by me, and it's because my legs and I are being pulled out sideways from the straight-line trajectory.

There's no rule saying it absolutely has to be this way, and for example, I can't juggle on a train with no walls, the effective 100 km/hr wind will blow my juggling balls far away as I throw them. If I didn't know what wind was, I would just say that there is a mysterious force on the balls that is reference-frame-dependent. But, coming up with that sort of theory is harder.

There is actually one more consideration, which is that the train needs to not be going way faster than trains usually go. So, this has to do with the curve of the earth underneath you, if the train starts moving tens of thousands of kilometers per hour, I will actually start to perceive gravity diminishing on this train, and when the train goes fast enough it will essentially be in orbit about the earth, and there will be zero gravity perceived in the train! So for this force I really would write an expression that was not reference-frame independent, called the Coriolis force term.

But the claim is, if the train keeps a uniform motion in a straight line, and the velocity is not too high and so forth: then juggling for me on the train is no different than juggling for me on the ground. This is just what I observe. The only thing that matters is the relative motion of the balls relative to my hands then, and not the absolute motion of the balls relative to Earth. And this is not a theoretical consideration but a simple observed fact, people juggle on trains all the time.

This observed fact was incorporated into Newton's laws of motion. Before Newton people often thought that forces immediately produced velocities, so that the land has some objective property of stillness and if something is moving across that stillness then you need an explanation in terms of a force, $F= mv$. So the inertia of an object was kind of its own force back in those days! Think about it: you kick a soccer ball, it is very easy to understand how it gets its initial motion, as you were kicking it. But then it keeps flying through the air and it does not immediately stop when your foot is not on it. So you have to invent these other strange “forces” and so forth, it gets really tricky.

Newton instead defined a force to be an acceleration, a change in velocity per unit time, $m(v_1-v_0)/(t_1-t_0).$ If we have a train, some new velocities $u_{0,1}=V+v_{0,1}$ where $V$ is the velocity of the train, notice that $$u_1-u_0=(V+v_1)-(V+v_0)=v_1-v_0,$$and what this means is, our definition of force is reference-frame invariant: it doesn't care whether we are on a train or the ground.

And that gets to the real point, which is that expressions that are not reference-frame invariant are much more complicated to deal with, because force wants to be reference-frame independent. So those sorts of forces are much harder to express in terms of potential energy functions, for example. And this also means that it's harder to write a Lagrangian for them and do all of our other classical mechanics trickery. The point is that this extra complexity has not been necessary to understand the physics of elementary particles, like it's not a part of the Standard Model, that sort of thing. When we are introducing the term like this Coriolis force, we usually think that it's because our coordinate systems failed to be inertial, rather than the conceptual difficulty of reference frame inequivalence.

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  • $\begingroup$ thank you so much for this answer. It really helped me. My main challenge seems to be in two directions 1) how problems in Physics are approached compared to my field (CS), and 2) I am struggling not with the complete idea of relative motion but with why the physics on of say two spaceships in relative motion to each other (from their respective frames) MUST be the same. I generally understand why the forces should be the same, but cannot understand why they should have exactly the same values. $\endgroup$ Aug 25, 2021 at 15:01
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    $\begingroup$ @DimitarDimitrov If you would like there is a book, by MIT’s Gerry Sussman who did half of the now famous CS “Structure and Interpretation of Computer Programs” lectures and textbook (watch these lectures if you haven't yet! They are worth every second of cheesy 80s goodness), a parallel work for physics called the “Structure and Interpretation of Classical Mechanics”. For free from MIT. Enjoy. $\endgroup$
    – CR Drost
    Aug 26, 2021 at 2:37
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    $\begingroup$ As for the rest, please note that there is no reason why it MUST be so directly. It is just that the theories where it is not so, are more complicated than we need. So, I can introduce a force term that depends on velocity. If I am doing biophysics, and I am concerned about bacteria moving around in the ocean, then I need this force term—such linear drag is ubiquitous for those microorganisms. At larger scales that decays to quadratic drag, which gets to an interesting point, it will be very hard for me to juggle on this train if it does not have walls! The wind will carry my balls away! $\endgroup$
    – CR Drost
    Aug 26, 2021 at 2:47
  • $\begingroup$ Really nice answer. $\endgroup$ Sep 2, 2021 at 3:45
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The point to bear in mind is that all motion is relative- all speeds must be defined relative to a point of reference. You, sitting at your laptop reading this, do not have a single fixed speed- you are simultaneously moving at all kinds of different speeds relative to different reference points. You are stationary, perhaps, relative to your chair. You are moving at perhaps ten metres per second relative to cars passing your house. You might be moving at a few hundred miles per hour relative to planes overhead. You are moving at a few thousand miles per hour relative to people on the other side of the world. You are even moving at speeds close to the speed of light relative to galaxies in the far distant universe. How can you say that the laws of physics you observe depend on your speed, when you don't have an absolute speed?

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  • $\begingroup$ Thank you for this @Marco Ocram. This helped me further improve how I think about relative motion. My main challenge was that I struggled with why should the physical forces have exactly the same measurements on two ships traveling relative to each other (from their respective frames) with different velocities. Feel free to add to this if you think there is anything that can help me improve my way of approaching it. Right now, it appears that my struggle with the why my not be the right focus. I was thinking from the perspective: Why must this be true? $\endgroup$ Aug 25, 2021 at 14:46
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There is one point where I'm not satisfied with some of the answers you received. Some appear to believe that the principle of relativity (PR) may be proven. For instance, @MarcoOcram writes:

How can you say that the laws of physics you observe depend on your speed when you don't have an absolute speed?

Now, one thing is the right statement that in order to speak of a velocity, you have before to define the frame of reference wrt that velocity is measured. Another is to state that there is no absolute velocity. This statement could be factually true or false, and only experiments can tell.

In the ether vision of the nineteenth century, there was an absolute velocity: that wrt ether. Einstein's PR simply declared that there is no ether: that even Maxwell's equations do hold in whatever inertial frame.

But he could have been wrong: some facts already known at his times spoke in favor of that general PR, many other came afterward. And this is the (experimental) foundation of the PR.

This answer also holds for @Martin and @Brondahl.


There is a second point that struck me.

Some responders attribute the PR to Einstein tout court; others attribute it to Newton; finally, others generically refer to it as a theorem in Newtonian mechanics.

No one has reported the true origin of the PR, which can be found in the Dialogue Concerning the Two Chief World Systems by Galileo Galilei (1632).

There is a most famous page of that book which runs very like to @thinking_squares' answer. The book is written in Italian, but you can easily find English translations, e.g. here.

Of course, neither Newton nor Galileo knew anything about electric and magnetic fields; let alone about electromagnetic waves. This is why Galileo speaks of flying birds, of jumping, of throwing balls...

Actually, once Newtonian mechanics is established, the PR easily follows as a theorem about motions. You have only to assume that all forces only depend on bodies' distances and relative velocities.

All that was well known at Einstein's times. But it was also thought that there was a preferred reference frame: that of ether, the medium wherein electromagnetic waves propagate. Then PR couldn't hold true for electromagnetic phenomena.

There was a problematic situation, with experiments perhaps contradicting that view. It was in such a tangle that Einstein had the courage to generalize Galileo's statement to all physical phenomena, and of deducing the necessary consequences thereof: first of all, the relative character of time (i.e. its dependence on the reference frame).


A personal note.

I felt obliged to the above notes since in my career as a physics professor I had the opportunity to teach relativity (special and general) for several years in a University located in the town of Pisa (Italy): Galileo's birthplace.

(The sketch in my profile depicts Galileo dropping from the top of the Leaning Tower two balls of very different weights, to show they reach ground together.)

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  • $\begingroup$ Thank you @Elio Fabri. These really helped. Also, I do find strange that Galileo is rarely mentioned. There is hope though, Jim Al-Khalili clearly calls out the origin of the relativity idea in Galileo's writing in his book "The World According To Physics". Hopefully that will help the uninitiated folks like me learn that fact and cary it forward. $\endgroup$ Aug 25, 2021 at 15:06
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From the point of experience

You tell that "I cannot understand why the two observers moving at constant velocity, which obviously differs between them in order for their motion to quality as relative motion, would observe the same physics." -- If you feel this way, you must be able to provide an example where the two observers observe different physics. Please share that example with us, it will help to guide you further.

From the point of axiomatics

The quote "the laws of physics—whatever they may be—must be absolutely identical for all observers undergoing constant-velocity motion" in the given context can also be understood as a requirement. We simply postulate that it is true, build a theory based on it and see if predictions of that theory matches our experiments.

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  • $\begingroup$ I am coming from a field where interactions within the models are a priori and there is a reason for each one... or something breaks. The way I was approaching the statement "the results they find will be identical" was by asking "why must they be? What will break if that is the case?". The "why must they be" in this case seems to be that if they were not, they would not agree with our evidence (experience). I am not used to think in the terms of hypothesizing an observed behavior to explain it as all behavior is predetermined (with some exception in Quantum Computing, etc.) in CS. $\endgroup$ Aug 25, 2021 at 15:17
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    $\begingroup$ @DimitarDimitrov, your first comment I think makes it more clear to me where you're coming from :) And those are very much questions that abstract mathematicians ask a lot. A lot of interesting maths is derived from mathematicians going "well ... we all know that reality isn't really like this ... but what would happen if it were? What would be the result?" $\endgroup$
    – Brondahl
    Aug 25, 2021 at 15:56
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    $\begingroup$ Sometimes the answer ends up with "you get paradoxes and contradictions", sometimes the answer is "meh ... you get a universe that's fine but a little different", and sometimes the answer is "Oh shit ... actually that explains some things we were confused about ... maybe this IS actually how reality works after all!" $\endgroup$
    – Brondahl
    Aug 25, 2021 at 15:56
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To get it out of the way, the answers are "yes", "yes", "no" according to the theory of special relativity.

I will venture a guess that the issue here is the interpretation of the phrase "the physics is the same". Does this mean that the observed values of all physical phenomena are the same? If it did, the statement would clearly be false: for example, in the classical scenario where Alice is on a moving train and Bob is on the platform, they will disagree on the speed of the train. Alice, who is moving with it, will measure the speed to be 0, while Bob will measure some positive value.

What does the phrase mean, then? Einstein said it best, but let me give an example.

Say Alice measures a certain electric field on the train. Bob may also make certain measurements, and he will find a different value for that electric field, as well as a small magnetic field (this can be calculated by looking at the Lorentz transformation of the electromagnetic tensor).

So, they will disagree about the value of the fields $\vec{E}$ and $\vec{B}$: however, the crucial point is that these fields will satisfy Maxwell's equations in both frames.

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    $\begingroup$ Yes, you are absolutely right. I wish I expressed my confusion better as I stated the question. So just to be clear, you are saying that that while the measurements may not be the same between the two reference frames, they will be consistent within each frame (satisfy Maxwell's equations) and thus produce equivalent physical experience. However, you are saying that there is still no condition for which the physics will differ between the frames. $\endgroup$ Aug 25, 2021 at 15:11
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I think something that can make this more intuitive is to start, not by thinking in terms of physics, but in terms of geometry. Think about lines on a plane and ask yourself if the geometry is the same for all of them. If we had no coordinate system would we be able to distinguish between the lines? You can think about moving at constant velocity in spacetime as moving in lines through spacetime, and you can ask yourself if there is an absolute coordinate system in spacetime.

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A little reminder may help your intuition: You are wondering, maybe, why a pilot in a jet experiences "the same physics" as you do. You are, after all, sitting still in your chair, while the pilot is hurtling through the air, driven by explosive forces to supersonic speeds.

Well. Are you stationary? It sure seems so. But at the equator you'd be moving at a speed of 42,000km/24h around Earth's axis (and the fighter pilot, flying west, could be standing still ...).

The Earth is orbiting the Sun at about 30km/s. The Sun orbits the galactic center at maybe 210km/s. The local group of which our galaxy is a part moves with hundreds of km/s towards the Virgo Cluster.

And still, if you watch the clicker-di-clack of Newton's cradle in the physics laboratory the laws of motion seem to be perfectly valid, right now, and a second later in a place that was deep vacuum the second before. You wouldn't even know you are moving — moving relative to the Earth axis, the Sun, the Milky Way and the Virgo Cluster — if you didn't have (radio) telescopes. That's all there is to it.

(Nitpickers may remark that with all the overlayed circular and otherwise accelerated motions we are not even in a proper inertial system; but the radii are all large enough that we don't run into our door frames, so for our purposes it's good enough.)

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Computer Science Shows the Existence of Reference Frames as a Natural Consequence of Grouping Systems

Since you say you come from a computer science background, it is worth noting that every simulation that allows for nesting moving objects creates reference frames. Let's look at this problem just like a computer simulation operating in a cardigan space. To make this as simple as possible, let's say your simulation is 2 dimensional. If you've ever built a game engine from the ground up, you know that all objects in your game space need as a very minimum a position, velocity, and acceleration for each dimension so that each simulation cycle can determine how far the object moves.

So, to say all things are governed by the same set of physics is like saying we are all created from the same object class:

class thing {  
  float posX; 
  float posY; 
  float velX; 
  float velY; 
  set(float pX, float pY, float vX, float vY){ // Set initial position and velocity. 
    posX = pX; 
    posY = pY; 
    velX = vX; 
    velY = vY;  
  }
  void move(float aX, float aY) { // Apply acceleration each cycle based on outside forces.
    velX += aX;
    velY += aY; 
    posX = velX;
    posY = velY;
    cout "("<< posX<<","<<posY<<")";
  }
}   

Every object in your cardigan space follows the same variables and move function meaning they are controlled by the same set of physics.

Now let's see what happens when we move two objects in parallel:

thing object1 = (0,0,10,0);
thing object2 = (5,0,10,0);
for (int i = 0; i < 3; i++) {
  object1.move;
  object2.move;
  cout "\n";
}

This will output:

(0,0)(5,0)
(10,0)(15,0)
(20,0)(25,0)

However, there is another way to think about two objects moving in parallel like this, and that is by nesting your objects recursively.

thing group = (0,0,10,0)
group.thing object1 = (0,0,0,0);
group.thing object2 = (5,0,0,0);
for (int i = 0; i < 3; i++) {
  group.move;
  cout "\n";
} 

Here you get the same output by applying the motion to the group of objects as you did when you were applying the motion individually. This means that group here is your reference frame because as far as object1 and object2 are concerned, they are both staying still; so, if you apply an acceleration to object1, its motion relative to object2 is the same as if they were staying still, or if they have an underlying acceleration/velocity being applied by the group.

So, basically, all reference frames are is a way you can mathematically take many particles, find 1 velocity and acceleration they all have in common, and treat their physics as a group instead of as individual objects. This works because the objects all inherit the same common set of inputs from the object, and then just add/subtract their own local movements to it.

The difference between a computer and physics is that with a computer, there is a guaranteed top-level object with a fixed coordinate system, and each recursive layer is a fixed part of your data structure, but in physics, there is not necessarily a fixed data structure as far as anyone can prove. We just fit our mathematic models that work in a simulation to what matches reality and this kind of recursive object structure seems to match reality.

So, how does this all factor into your question? From the CS perspective, a reference frame is just a data model that recursively groups objects to simplify your math. If I were to take the above class and add a 3rd dimension, energy transfer equations, and all the other weird stuff that makes up real world physics, you would continue to see the same basic principles at work with one notable difference: Relativity basically says that you do not need a top level object, but that you can take any object and re-arrange everything to make that object work like the a top level object.

thing group = (0,0,10,0)
group.thing object1 = (0,0,0,0);
group.thing object2 = (5,0,0,0);

gives the same output as

thing group = (0,0,0,0);
group.thing object1 = (0,0,10,0);
group.thing object2 = (5,0,10,0);

or as

thing group = (5,0,0,0);
group.thing object1 = (-5,0,10,0);
group.thing object2 = (0,0,10,0);

So, this means to do any physics at all, you don't need to understand the top level frame of the universe (if there is one), you can just make all frames relative to any object of interest you want, and it works because every object follows the same functions no matter how you partition it. There are certain forces that are so darn small inside of what we would consider a reference frame that it looks like they don't apply, but even a slug moving along a leaf experiences time dilation; even a feather drifting in the breeze in pulling the Earth with its gravity.

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